One representation for spinors, four-vectors and six-vectors

The group SL(2,$\mathbb{C}$) is a suitable representation for transforming Weyl spinors and two index objects equivalent to four-vectors. In this sense, the electromagnetic four-potential also transforms under the group SL(2,$\mathbb{C}$). But, second order tensors such as the Faraday tensor  does not transform under this group. However, there is an alternative way of describing the electric and magnetic fields as a complex six-dimensional vector, namely the Riemann-Silberstein vector. In this note we show that it is possible to transform Riemann-Silberstein vector with the elements of the group SL(2,$\mathbb{C}$).